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Theorem iota4an 4962
Description: Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)
Assertion
Ref Expression
iota4an  |-  ( E! x ( ph  /\  ps )  ->  [. ( iota x ( ph  /\  ps ) )  /  x ]. ph )

Proof of Theorem iota4an
StepHypRef Expression
1 iota4 4961 . 2  |-  ( E! x ( ph  /\  ps )  ->  [. ( iota x ( ph  /\  ps ) )  /  x ]. ( ph  /\  ps ) )
2 euiotaex 4959 . . . 4  |-  ( E! x ( ph  /\  ps )  ->  ( iota
x ( ph  /\  ps ) )  e.  _V )
3 simpl 107 . . . . 5  |-  ( (
ph  /\  ps )  ->  ph )
43sbcth 2842 . . . 4  |-  ( ( iota x ( ph  /\ 
ps ) )  e. 
_V  ->  [. ( iota x
( ph  /\  ps )
)  /  x ]. ( ( ph  /\  ps )  ->  ph )
)
52, 4syl 14 . . 3  |-  ( E! x ( ph  /\  ps )  ->  [. ( iota x ( ph  /\  ps ) )  /  x ]. ( ( ph  /\  ps )  ->  ph )
)
6 sbcimg 2869 . . . 4  |-  ( ( iota x ( ph  /\ 
ps ) )  e. 
_V  ->  ( [. ( iota x ( ph  /\  ps ) )  /  x ]. ( ( ph  /\  ps )  ->  ph )  <->  (
[. ( iota x
( ph  /\  ps )
)  /  x ]. ( ph  /\  ps )  ->  [. ( iota x
( ph  /\  ps )
)  /  x ]. ph ) ) )
72, 6syl 14 . . 3  |-  ( E! x ( ph  /\  ps )  ->  ( [. ( iota x ( ph  /\ 
ps ) )  /  x ]. ( ( ph  /\ 
ps )  ->  ph )  <->  (
[. ( iota x
( ph  /\  ps )
)  /  x ]. ( ph  /\  ps )  ->  [. ( iota x
( ph  /\  ps )
)  /  x ]. ph ) ) )
85, 7mpbid 145 . 2  |-  ( E! x ( ph  /\  ps )  ->  ( [. ( iota x ( ph  /\ 
ps ) )  /  x ]. ( ph  /\  ps )  ->  [. ( iota x ( ph  /\  ps ) )  /  x ]. ph ) )
91, 8mpd 13 1  |-  ( E! x ( ph  /\  ps )  ->  [. ( iota x ( ph  /\  ps ) )  /  x ]. ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    e. wcel 1436   E!weu 1945   _Vcvv 2615   [.wsbc 2829   iotacio 4941
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-rex 2361  df-v 2617  df-sbc 2830  df-un 2992  df-sn 3437  df-pr 3438  df-uni 3637  df-iota 4943
This theorem is referenced by: (None)
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